OK,yah...I can understand that.However,if they're not US citizens and don't have a Green Card (which would mean either that they were in the US illegally or on some kind of temporary visa) you'd have to deport them to their country of citizenship.
I've seen it written that there are some who enter the US illegally who,when caught,refuse to present any papers (passport,etc) and also refuse to identify their nationality.IIRC our courts have ruled that you can't deport such people and you can't detain them for any length of time.That means they get to stay.Don't know if the same is true Up North.
http://montreal.ctvnews.ca/troops-deployed-to-set-up-tents-at-border-old-royal-vic-to-shelter-asylum-seekers-1.3538567
If you turn them around at the border, they end up back in the US.
You are right about the rest. They arrive and claim to be without document, hard to deal with. Actually that was the origin of the explicative "WOP", With Out Papers.